Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. In-Class Activities: Check Homework Reading Quiz Applications Types of Internal Forces Steps for Determining Internal Forces Concept Quiz Group Problem Solving Attention Quiz Today’s Objective: Students will be able to: 1.Use the method of sections for determining internal forces in 2-D load cases. INTERNAL FORCES

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. 1. In a multiforce member, the member is generally subjected to an internal _________. A) Normal force B) Shear force C) Bending moment D) All of the above. 2. In mechanics, the force component V acting tangent to, or along the face of, the section is called the _________. A) Axial force B) Shear force C) Normal force D) Bending moment READING QUIZ

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. Why are the beams tapered? Internal forces are important in making such a design decision. In this lesson, you will learn about these forces and how to determine them. Beams are structural members designed to support loads applied perpendicularly to their axes. Beams can be used to support the span of bridges. They are often thicker at the supports than at the center of the span. APPLICATIONS

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. Usually such columns are wider/thicker at the bottom than at the top. Why? A fixed column supports these rectangular billboards. APPLICATIONS (continued)

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. Why might have this been done? The shop crane is used to move heavy machine tools around the shop. The picture shows that an additional frame around the joint is added. APPLICATIONS (continued)

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. Then we need to cut the beam at B and draw a FBD of one of the halves of the beam. This FBD will include the internal forces acting at B. Finally, we need to solve for these unknowns using the E-of-E. For example, we want to determine the internal forces acting on the cross section at B. But, first, we first need to determine the support reactions. B The design of any structural member requires finding the forces acting within the member to make sure the material can resist those loads. B INTERNAL FORCES

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. The loads on the left and right sides of the section at B are equal in magnitude but opposite in direction. This is because when the two sides are reconnected, the net loads are zero at the section. In two-dimensional cases, typical internal loads are normal or axial forces (N, acting perpendicular to the section), shear forces (V, acting along the surface), and the bending moment (M). INTERNAL FORCES (continued)

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. 1. Take an imaginary cut at the place where you need to determine the internal forces. Then, decide which resulting section or piece will be easier to analyze. 2. If necessary, determine any support reactions or joint forces you need by drawing a FBD of the entire structure and solving for the unknown reactions. 3. Draw a FBD of the piece of the structure you’ve decided to analyze. Remember to show the N, V, and M loads at the “cut” surface. 4. Apply the E-of-E to the FBD (drawn in step 3) and solve for the unknown internal loads. STEPS FOR DETERMINING INTERNAL FORCES

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. Solution 1.Plan on taking the imaginary cut at C. It will be easier to work with the right section (from the cut at C to point B) since the geometry is simpler and there are no external loads. Given: The loading on the beam. Find:The internal forces at point C. Plan:Follow the procedure!! EXAMPLE

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. Applying the E-of-E to this FBD, we get  +  F x = B x = 0; +  M A = − B y ( 9 ) + 18 ( 3 ) = 0 ; B y = 6 kip 2. We need to determine B y. Use a FBD of the entire frame and solve the E-of-E for B y. BxBx 3 ft9 ft AyAy ByBy 18 kip 3 ft FBD of the entire beam: EXAMPLE (continued)

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. 3. Now draw a FBD of the right section. Assume directions for V C, N C and M C. 6 kip VCVC MCMC NCNC 4.5 ft C B 4. Applying the E-of-E to this FBD, we get  +  F x = N C = 0; N C = 0  +  F y = – V C – 6 = 0; V C = – 6 kip +  M C = – 6 (4.5) – M C = 0 ; M C = – 27 kip ft EXAMPLE (continued)

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. 1. A column is loaded with a vertical 100 N force. At which sections are the internal loads the same? A) P, Q, and R B) P and Q C) Q and R D) None of the above. P Q R 100 N 2. A column is loaded with a horizontal 100 N force. At which section are the internal loads largest? A) P B) Q C) R D) S P Q R 100 N S CONCEPT QUIZ

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. GROUP PROBLEM SOLVING I Given: The loading on the beam. Find:The internal forces at point C. Plan:Follow the procedure!! Solution: 1. Plan on taking the imaginary cut at C. It will be easier to work with the left section (point A to the cut at C) since the geometry is simpler.

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. GROUP PROBLEM SOLVING I (continued) 2. First, we need to determine A x and A y using a FBD of the entire frame. Applying the E-of-E to this FBD, we get  +  F x = A x = 0 ; A x = – 400 N +  M B = – A y (5) – 400 (1.2) = 0 ; A y = – 96 N 400 N ByBy AxAx AyAy Free Body Diagram

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. GROUP PROBLEM SOLVING I (continued) 3. Now draw a FBD of the left section. Assume directions for V C, N C and M C as shown. 4. Applying the E-of-E to this FBD, we get  +  F x = N C – 400 = 0; N C = 400 N  +  F y = – V C – 96 = 0; V C = – 96N +  M C = 96 (1.5) + M C = 0 ; M C = -144 N m 96 N VCVC MCMC NCNC 1.5 m A 400 N C

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. Solution: 1. Make an imaginary cut at C. Why there? Which section will you pick to analyze via the FBD? Given: The loading on the beam. Find:The internal forces at point C. Plan: Follow the procedure!! Why will it be easier to work with segment AC? GROUP PROBLEM SOLVING II

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. +  M A = T ( 2.5 ) − 1800 (6) = 0 ; T = 4320 lb  +  F x = A x − 4320 = 0 ; A x = 4320 lb  +  F y = A y − 1800 = 0 ; A y = 1800 lb 2. Determine the reactions at A, using a FBD and the E- of-E for the entire frame. GROUP PROBLEM SOLVING II (continued) T AxAx AyAy 1800 lb 6 ft Free Body Diagram

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. 3. A FBD of section AC is shown below. GROUP PROBLEM SOLVING II (continued) VCVC MCMC NCNC 1.5 ft A C 450 lb 1.5 ft FBD of Section AC 4320 lb 1800 lb 4. Applying the E-of-E to the FBD, we get  +  F x = N C = 0 ; N C = – 4320 lb  +  F y = 1800 – 450 – V C = 0 ; V C = 1350 lb +  M C = – 1800 (3) (1.5) + M C = 0 ; M C = 4725 lb  ft

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved. 2. A column is loaded with a horizontal 100 N force. At which section are the internal loads the lowest? A) P B) Q C) R D) S P Q R 100N S 1. Determine the magnitude of the internal loads (normal, shear, and bending moment) at point C. A) (100 N, 80 N, 80 N m) B) (100 N, 80 N, 40 N m) C) (80 N, 100 N, 40 N m) D) (80 N, 100 N, 0 N m ) C 0.5m 1 m 80 N 100 N ATTENTION QUIZ

Statics, Fourteenth Edition R.C. Hibbeler Copyright ©2016 by Pearson Education, Inc. All rights reserved.